1. Introduction Let E be an elliptic curve over Q. We recall that the Tate-Shafarevich group of E/Q is defined by

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1 Bull. Korean Math. Soc. 50 (2013), No. 2, pp ON THE p-primary PART OF TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER Q WHEN p IS SUPERSINGULAR Dohyeong Kim Abstract. Let E be an elliptic curve over Q and p be a prime of good supersingular reduction for E. AlthoughtheIwasawatheoryofE over the cyclotomic Z p-extension of Q is well known to be fundamentally different from the case of good ordinary reduction at p, weareabletocombinethe method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the number of copies of Q p/z p occurring in the p-primary part of the Tate-Shafarevich group of E over Q. 1. Introduction Let E be an elliptic curve over Q. We recall that the Tate-Shafarevich group of E/Q is defined by ( X(E/Q) =Ker H 1 ( Q,E(Q) ) H 1 ( Q v,e(q v ) )), v where v runs over all places of Q, andq v is the completion of Q at v. It is well-known that the p-primary subgroup of X(E/Q) has a finite Z p -corank, and we denote this corank by t p. It is conjectured that t p = 0 for every prime p, but this is unknown when the complex L-function has a zero of order at least 2 at s = 1. We say a prime p is good ordinary (resp. good supersingular) if E has good ordinary reduction (resp. good supersingular reduction) atp. The aim of present paper is to compute an upper bound for t p for all good supersingular primes p. For the case of good ordinary primes, see the author s previous paper [4]. If E has complex multiplication and p is ordinary, then a better bound is obtained in [2]. Suppose from now on that p is a prime of good supersingular reduction for E. We recall that if F is a finite extension of Q, thenthe(p -)Selmer group Received March 9, 2011; Revised March 8, Mathematics Subject Classification. 11G05. Key words and phrases. Iwasawa theory, supersingular prime, elliptic curves, Tate- Shafarevich group. 407 c 2013 The Korean Mathematical Society

2 408 DOHYEONG KIM of E over F is defined as ( Sel(E/F) := Ker H 1 (F, E[p ]) v ) H 1 (F v,e), where v runs over all places of F,andF v is the completion of F at v, and E[p ] is the Galois module of p-power division points in E(Q). Let Q cyc be the cyclotomic Z p -extension of Q. Put Sel(E/Q cyc ) = lim Sel(E/F), where F runs over the finite extensions of Q contained in Q cyc and the inductive limit is taken relative to the restriction maps. Let Γ := Gal(Q cyc /Q) anddefine the Iwasawa algebra Λ(Γ) by Λ(Γ) = lim Z p [Γ/U], where U runs over the open subgroups of Γ. The natural action of Γ on H 1 (Q cyc,e[p ]) gives rise to a continuous action of Γ on Sel(E/Q cyc ), which extends to an action of the whole Iwasawa algebra Λ(Γ). For an abelian group A, denote by A the Pontryagin dual Hom(A, Q p /Z p )ofa. We write the Pontryagin dual of a Selmer group by X(E/F) := Sel(E/F) := Hom(Sel(E/F), Q p /Z p ). Our previous method used in [4] heavily depends on the important theoremthat X(E/Q cyc ) is a torsion module over Λ(Γ) when p is a good ordinary prime. In contrast, when p is a good supersingular prime, which is the case of our interest, it is known that X(E/Q cyc ) is not a torsion Λ(Γ)-module. Nevertheless, we can use the ingenius idea of Kobayashi [5] of considering modified Selmer groups, the duals of which are torsion over Λ(Γ). Combined with our earlier method in [4], we can compute an upper bound for t p for supersingular primes p. Let δ = 1 when the sign is + and δ = 0 otherwise. Our main results are: Theorem 1.1. Let L ± p (E,α,T) be Pollack s modified p-adic L-functions. Let T m± p (E) be the exact power of T dividing L ± p (E,α,T). Then for a constant C = C(E) which depends on E but not on p, wehavem ± p (E) Cp8+δ for all good supersingular primes p. The definition of the power series L ± p (E,α,T) in Z p [[T ]] is given in Section 2. As a corollary, we prove: Corollary 1.2. Let g E be the rank of E(Q). For all primes p where E has good supersingular reduction, we have t p Cp 8 g E. The proofs of Theorem 1.1 and Corollary 1.2 are similar to those in our earlier paper [4]. However, there are additional estimations involving the modified p-adic logarithm map. We remark that in the case when E has complex multiplication, an alternative method used in [2] to obtain a better bound for t p for sufficiently large ordinary primes does not yet generalize to supersingular primes.

3 ON THE p-primary PART OF TATE-SHAFAREVICH GROUP Iwawawa theory for a supersingular prime In this section, we recall the Iwasawa theory of E at a supersingular prime p. On the algebraic side, we will define the modified Selmer groups following Kobayashi [5], while on the analytic side we will use the theory of Pollack [8] who constructed the modified p-adic L-functions L ± p (E,α,T). We fist begin with the modified Selmer groups. For a each n 0, let F n = Q(µ p n+1) and let F 1 = Q. LetK n be the completion of F n at the unique prime of F n lying above p. Also let Tr n,m : E(K n ) E(K m )bethetracemapwhenn m. Plainly, we have E(K m ) E(K n )whenn m. Definition 2.1. We define the n-th modified Selmer groups as ( Sel ± (E/F n ) := Ker Sel(E/F n ) H1 (K n,e[p ) ]) E ±, (K n ) Q p /Z p where E + (K n ) := {P E(K n ) Tr n,m+1 (P ) E(K m )forevenm (0 m<n)} and E (K n ) := {P E(K n ) Tr n,m+1 (P ) E(K m )foroddm (0 m<n)}. We call Sel + (E/F n )(resp. Sel (E/F n )) the even (resp. odd) Selmer group over F n. We adopt the convention that E ± (K 1 )=E(Q p ), whence it is clear that we have Sel ± (E/Q) = Sel(E/Q). One sees easily that the restriction map from Sel(E/F n ) to Sel(E/F n+1 )maps Sel ± (E/F n )tosel ± (E/F n+1 )andsowecandefine Sel ± (E/Q cyc ) = lim Sel ± (E/F n ). Also, it is proven in Lemma 9.1 of [5] that the map Sel ± (E/F n ) Sel ± (E/Q cyc ) is injective. As usual Selmer groups, the modified Selmer groups over Q cyc are also modules over Λ(Γ). However, the modified Selmer groups are crucially different from usual Selmer groups in that they are torsion Λ(Γ)-modules. Kobayashi proved (Theorem 1.2 in [5]): Theorem 2.2. The modified dual Selmer groups X ± (E/Q cyc ) := Sel ± (E/Q cyc ) are finitely generated torsion Λ(Γ)-modules. Let R = Z p [[T ]] be the power series ring in a variable T with coefficients in Z p. Fix a topological generator γ of Γ. The Z p -algebra homomorphism Λ(Γ) R sending γ to T + 1 is an isomorphism. Using this isomorphism, elements in Λ(Γ) will be considered as power series in the variable T. By the

4 410 DOHYEONG KIM structure theorem of finitely generated torsion R-modules, there exists non-zero elements G ± j (T ) R, finite R-modules D± and exact sequences 0 k ± j=1 R/G± j (T )R X ± (E/Q cyc ) D ± 0. The products G ± (T )= k ± j=1 G± j (T ) are then well-defined up to multiplication by a unit in R, andg ± (T ) are called the characteristic power series of X ± (E/Q cyc ). The Iwasawa main conjecture is an assertion that the above characteristic power series have analytic descriptions via the special values of the complex L-function, which we now explain. Let N E be the conductor of E. Since E is modular, there is a primitive normalized Hecke eigenform f(q) = k=1 a kq k of level N E weight 2 such that a p = p +1 Ẽp(F p ) for all primes p prime to N E. For a Dirichlet character χ whose conductor is prime to N E, the twisted L-function attached to f defined by χ(n)a n L(f,χ,s) =L(E,χ,s)= n s has analytic continuation to the entire complex plane. Let α and β be the roots of the equation X 2 a p X + p in Q. We fix embeddings of Q into C and Q p. Using these embeddings, α and β will be considered as elements of C and Q p. And let Ω + E be the least positive real period of the Nèron differential on the minimal Weierstrass model of E. Mazur and Swinnerton-Dyer [6] proved that we can p-adically interpolate values L(E,χ,1) as χ varies among Dirichlet characters of p-power conductor and order. We now recall a special form of such interpolation property (See Section 14 of [7]) expressed in terms of a power series. For a character χ of conductor p r+1,wedefinethegausssumas p r+1 i=1 ζ i χ(i), where ζ is a primitive p r+1 -th root of unity. Theorem 2.3. Fix a global minimal generalized Weierstrass equation for E. Let p be a prime of good supersingular reduction for E. Letχ be a non-trivial Dirichlet character of conductor p r+1 and order p r.thenthereexistsaunique power series L p (E,α,T) Q p (α)[[t ]] with the following properties. Firstly, L p (E,α,ζ 1) converges for every p- power roots of unity ζ. Secondly, thispowerseriesinterpolatethecomplex L-values in the sense that (1) L p (E,α,χ(γ) 1) = pr+1 L(E,χ, 1) Ω + E αr+1 τ(χ) for every nontrivial Dirichlet characters χ of p-power order and conductor. n=1

5 ON THE p-primary PART OF TATE-SHAFAREVICH GROUP 411 Note that for a supersingular prime p, thep-adic L-function L p (E,α,T) is not in Z p [[T ]]. As an analytic counterpart of the modified Selmer groups which are torsion over Λ(Γ), Pollack [8] introduced the following p-adic logarithmic functions to define the corresponding modified p-adic L-functions which actually lie in Z p [[T ]]. Let Φ n (T )= p 1 t=0 T tpn 1 be the p n -th cyclotomic polynomial. Lemma 2.4. The products log + p (T ) := 1 p log p (T ) := 1 p ( ) Φ2n (1 + T ), p n=1 ( ) Φ2n 1 (1 + T ) n=1 converge and define power series in Q p [[T ]] which are convergent on the open unit disc. The zeros of log + p (resp., log p ) are precisely ζ 2n 1(resp., ζ 2n 1 1) for a p n -th root of unity ζ n with n>0, andtheseareallsimplezeros. Proof. See Lemma 4.1 of [8]. We remark that the above definition is the weight 2 case of the construction in Lemma 4.1 [8] where modular forms of weight k 2 are considered. Furthermore, we can write explicit interpolation properties of them. Lemma 2.5 ([8], Lemma 4.7). Let ζ r be a primitive p r+1 -th root of unity. We write Φ j for the p j -th cyclotomic polynomial. Then we have 0 when 2 r, log + (r 1)/2 p (ζ r 1) = p (r+1)/2 Φ 2j (ζ r ) when 2 r log p (ζ r 1) = r/2 p r/2 1 j=1 j=1 0 when 2 r. p Φ 2j 1 (ζ r ) when 2 r, Proof. See Lemma 4.7 of [8]. The role of log ± p is analogous to what a gamma function does to the Riemann zeta function. It exactly cancels out the trivial zeros and we are interested the remaining ones. In a more precise terms, we have: Theorem 2.6 ([8], Theorem 5.6). Under the assumption that p is odd and a p =0,wehave (2) L p (E,α,T)=L + p (E,α,T) log + p (T )+L p (E,α,T) log p (T )α, with L ± p (T ) Z p [[T ]].

6 412 DOHYEONG KIM Note that the assumption that a p = 0 is harmless for us since we can ignore finitely many primes when we prove Theorem 1.1, and a p =0foreveryp>5 due to the Hasse s bound a p 2 p. The Iwasawa main conjecture for the modified Selmer groups is the equality of L ± p (T )andg± (T ) up to multiplication by unit in Z p [[T ]]. Kobayashi proved the following in Theorem 1.3 [5] which amounts to one divisibility of the main conjecture, using Kato s Euler system. Theorem 2.7. Let G ± (T ) be a characteristic power series of X ± (E/Q cyc ). Then G ± (T ) divides L ± p (T ). 3. Proofs of the main results Let X(E/Q)[p ]bethep-primary part of X(E/Q). Recall that the Tate- Shafarevich group and the Selmer groups are related by the short exact sequence (3) 0 E(Q) Q p /Z p Sel(E/Q) X(E/Q)[p ] 0. If E(Q) is of rank g E,andX(E/Q)hasZ p -rank h p,thent p = h p g E.Therefore we obtain an upper bound for t p from that for h p. Thus it suffices to relate h p and the order of vanishing of L ± p (E,α,T). We first prove some preliminary results for the proof of the main theorem. Let ϕ(m) be the Euler totient function. Lemma 3.1. Let k be an integer such that 0 <k<n.thenwehave (4) Φ k (ζ n ) p = p ϕ(pn k ) 1 +ϕ(p n k+1 ) 1. In particular, we have log ± p (ζ r 1) p = p c(r,p), where c(r, p) is a function which is very small in the sense that for each fixed r>1, 0 <c(r, p) <rfor all but finitely many primes p. Proof. We prove it using the explicit cyclotomic polynomials. Write Φ k (T )= T pk 1 T pk 1 1. Then ζn pk and ζpk 1 n are respectively primitive p n k -th and p n k+1 -th roots of unity. By the formula ζ r 1 p = p ϕ(pr ) 1, we get equation (4). The second part comes from direct calculation using Lemma 2.5. Indeed, for odd positive integer r log + p (ζ r 1) p = p (r+1)/2 (r 1)/2 j=1 Φ 2j (ζ r ) = p r ( ϕ(p r 2j ) 1 +ϕ(p r 2j+1 ) 1 ), p

7 ON THE p-primary PART OF TATE-SHAFAREVICH GROUP 413 where the summation runs over j =1, 2,...,(r 1)/2 in the second line. For a fixed r, the terms in the summation in the second line converges to zero, thus c(r, p) is a decreasing function which converges to r+1 2 as p goes to infinity. In particular, for all but finitely many primes p, wehave0<c(r, p) <r. For an even positive r, a similar calculation shows c(r, p) is bounded by r as p goes to infinity. For a Dirichlet character χ of conductor p r and order p r 1 with r 2, precisely one of the two values log ± p (χ(γ) 1) vanishes according to Lemma 2.5. Therefore, by (2), nonvanishing of twisted L-values will give nonvanishing of modified p-adic L-functions. Proposition 3.2. Let T m± p (E) be the exact powers of T dividing L ± p (E,α,T). Then we have the inequality Proof. Consider the restriction map h p m ± p (E). res: Sel(E/Q) Sel ± (E/Q cyc ). As shown in the proof of Proposition 2 in [4], the image of res is fixed by Γ and the Pontryagin dual of Ker(res) is finite. Therefore, taking Pontryagin dual of res, we obtain Γ-equivariant surjections X ± (E/Q cyc ) Γ X(E/Q) with finite cokernels. This completes the proof. Lemma 3.3. There exists a constant c E which only depends on E but not on p such that α r+1 c E L ± p (E,α,χ(γ) 1) is an algebraic integer in Q(χ). Proof. It is shown in Proposition 1 in Section 3 of [4] that there exists a constant c E which only depends on E but not on p such that α r+1 c E L p (E,α,χ(γ) 1) is an algebraic integer in Q(χ). Although we made the assumption that p is ordinary in [4], the proof therein does not use it. Finally, we can prove the main theorem. From now on, we assume that the conductor of χ is p 9+δ and the order of χ is p 8+δ,whereδ is 0 or 1. Theorem 3.4. Given an elliptic curve E defined over Q, there is a constant C which depends on E but not on p, suchthatm ± p (E) Cp8+δ for all supersingular primes p. Hereδ =1when the sign is + and δ =0otherwise. Proof. It suffices to prove the assertion for sufficiently large p since we can adjust C to cover the remaining finitely many cases. Let χ be a Dirichlet character of conductor p 9+δ and order p 8+δ. By Theorem 3 of [1], we know the nonvanishing L(E,χ,1) 0

8 414 DOHYEONG KIM for sufficiently large primes p. Now let x = α 9+δ c E L p (E,α,χ(γ) 1) and y = x σ. Then y belongs to Z and is nonzero. formula (5) a 1 p σ Gal(Q(χ)/Q) We are going to apply the following a to y, which is true for all nonzero integers a. ϕ(p 8+δ ) elements. Firstly, we have Note that Gal(Q(χ)/Q) has x σ p = α 9+δ c E L ± p (E,α,χ(γ) σ 1) log ± p (χ(γ) σ 1) p = p 9+δ 2 ce L ± p (E,α,χ(γ) σ 1) log ± p (χ(γ) σ 1) p = p 9+δ 2 ce p L ± p (E,α,χ(γ) σ 1) p p c(8+δ,p) p 9+δ 2 ce p χ(γ) σ 1 m± p (E) p p c(8+δ,p). In the first line to the second, we used Lemma 2.5, (2) and (1). From the first line to the second, we used the fact that α p = p 1/2 for supersingular primes p. From the second line to the third, we used Lemma 3.1. The argument for the inequality between the third and forth lines is the following. Recall that by the definition of m ± p (E) we can write L ± p (E,α,T) =T m± p (E) H(T )for some H(T ) R. Then H(χ(γ) σ 1) p 1, whence L ± p (E,α,χ(γ)σ 1) p χ(γ) σ 1 p. Applying equation (5) to y and taking the logarithm with base p (not the p-adic logarithm), we obtain (6) log p y 1 p = m ± p (E) ϕ(p8+δ ) ( 1 2 (9 + δ) + log p c E p + c(8 + δ, p) On the other hand, using (1) and the analytic bounds for complex L-functions, we can show that (7) x σ C 1 p 9+δ for sufficiently large primes p where C 1 is a constant depending on E but not on p. We give the argument for (7) briefly here. It is well known that the absolute value of the Gauss sum is τ(χ) = p 9+δ. Thus (1) can be written as x σ = c E p 9+δ L(E,χ, 1) (8) Ω + E τ(χ) (9) = p 9+δ 2 c E L(E,χ, 1) Ω + E Now (7) would follow from an upper bound of the form (10) L(E,χ, 1) p 9+δ 2. ).

9 ON THE p-primary PART OF TATE-SHAFAREVICH GROUP 415 since Ω + E and c E are independent of p. In fact, subconvexity bounds such as Theorem 3 of [3] shows better result then (10). However, these better estimations will hardly improve our final result so we use (10). The definition of y together with (7) implies that (11) y ( C 1 p 9+δ) ϕ(p 8+δ ). Apply (5) to y, then (6) and (11) imply that ϕ(p 8+δ ) ( log p C 1 +9+δ ) ( ) (12) 9+δ m ± p (E) ϕ(p8+δ ) + log 2 p c E p + c(8 + δ, p). Rewriting (12) as an upper bound for m ± p (E), we obtain m ± p (E) ϕ(p 8+δ ) (log p C ) (13) (9 + δ)+c(8 + δ, p) + log p c E p. By Lemma 3.1, c(8+δ, p) is an bounded by 8+δ, and log p c E 1 p is independent of p. Also we have ϕ(p 8+δ ) p 8+δ. Thus, for sufficiently large p, (13)becomes ( m ± p (E) p8+δ log p C δ ) (14) +(8+δ). 2 The assertion of the theorem follows if we take C = log p C (9 + δ)+(8+ δ) + log p c E p. Corollary 3.5. We have t p Cp 8 g E. Proof. Consider the Pontryagin dual of the short exact sequence (3). By Proposition 3.2 and Theorem 3.4, we have h p Cp 8+δ. Since Z p -rank is additive in exact sequences we also have h p = g E + t p. Taking δ = 0, we obtain the assertion. References [1] G. Chinta, Analytic ranks of elliptic curves over cyclotomic fields, J.ReineAngew.Math. 544 (2002), [2] J. Coates, Z. Liang, and R. Sujatha, The Tate-Shafarevich group for elliptic curves with complex multiplication II, MilanJ. Math. 78 (2010), no. 2, [3] W. Duke, J. B. Friedlander, and H. Iwaniec, Bounds for automorphic L-functions. II, Invent. Math. 115 (1994), no. 2, [4] D. Kim, On the Tate-Shafarevich group of elliptic curves over Q, Bull.KoreanMath. Soc. 49 (2012), no. 1, [5] S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent.Math. 152 (2003), no. 1, [6] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil Cuves, Invent.Math.25 (1974), [7] B. Mazur, J. Tate, and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, [8] R. Pollack, On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), no. 3,

10 416 DOHYEONG KIM Department of Mathematics Pohang University of Science and Technology Pohang , Korea address: